How do I change the style of emoji in SwiftKey?

It's not currently possible to change the emoji style within SwiftKey itself. SwiftKey uses the standard Android/iOS emoji font, which means that the emoji style you see within SwiftKey (emoji panel/emoji predictions) is dependent on your OS or the app you are using.

However, in most messaging apps you can choose which emoji style you see once you have input an emoji or sent/received a message.

For more information on why emoji look different within different apps, please see this article.